The Hamiltonicity Of Some Graphs

All graphs considered in this paper are finite,simple and undirected. Let G=(V(G). E(G)) be a graph .where V(G) and E(G) denote the vertex set and edge set and set of G respectively.For {v1,v2...vn} (?) V(G),we use G[v1,v2,...vn] to denote the subgraph ofA path(eircle) is called hamilton path(circle) if it pass every vertex of G just one time.A graph G is called [s, t]-graph,if there are. at least t edges in every induced subgraphs of s vertexs.Let P=v1v2...vr is a (v1. vr.)-path,vi,vj ∈(P) (i < j).we use v, Pv j and v1 Pvi to denote the subgraph vivi+1...vj and vjvj-1...v1 of P respectively.We use vi+l,vi-l to denotes the vertex vi+l and vi-l, vi+1,vi-1 is also denoted byri+. ri-.H1. H2 is the subgraph of G. Let NH1(H2) = {v ∈V(H1)) | (?)u ∈ V(H2). such that uv∈ E{G)}. NG(H) can be denoted to N(H) simply.V({vi}) can be denoted to N(v1) simply.Let x∈V(G),S (?) V(G) and H is the subgraph of G,G[S] denoted the induced subgraph of S. And G[S] can be denoted to [S] simply.N(x) denotes the adjcent vertex set of x which is in G.N[x] = N(x) ∪ {x}NH (x)- N(x)∩V(H).If x,y ∈ V(G),d(x,y) is the distance in G from x to y.For u,v∈ C.let uCv denote the consecutive vertices on C from u to v in the direction. The same vertices,in reverse order .are given by vCu.If u is on C, then the predecessor, successor,next predeceddor and next successor of u along the orientation of C are denoted by u-,u+,u--,and u++.respectively.G1 and G2 is the subgraph of G,G1△G2 is definded to the subgraph of G which is induced by E(G1) ∪ E(G2) - (E(G1)∩ E(G2)).A graph G is triangularly connected if for every pair of edges e1,e2 ∈ E(G).G has a sequende of 3-cycles C1, C2, ..., Cl such that e1∈ C1, e2 ∈ Cl. and such thatE(Ci)∩E(Ci+1)≠(?) (1≤i≤l-1).A graph G is pancyclic if for each integer k,3 ≤k≤|VG|.G has a k-cycle; G is vertex pancyclic if for each vertex v ∈ V(G),and for each integer k with 3≤k≤ |VG|, G has a A;-cycle Ck such that v ∈ V(Ck).In this paper we have the following theorems.Theorem 2.2.1 G is a [4,2]-graph,then(a)G is connected iff' G(?)k1,3 or G has the Hamilton path.(b)G is 2-connected iff G(?)K2,3 or G(?)Kk,1,3 or G has the Hamilton cyccle.Theorem 2.2.2 G is "2-connected [5,2]-graph .then G(?)K2,4 or G has Hamiltonpath.Theorem 2.2.3 G is k-conncted [k+2.k]-graph.then G(?)Kk-1VGk or G has Hamilton cycle.Theorem 3.2 G is triangularly connected qausi-claw-free graph, and |E(G)| ≥ 3.then G is pancyclic .Theorem 4.2 G is a 4-connected K1,4 - restricted graph of order n,n≥ -12 and n ≤ 6δ 12. then G is hamiltonian.